Misplaced Pages

Fraňková–Helly selection theorem

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
(Redirected from Fraňková-Helly selection theorem) On convergent subsequences of regulated functions

In mathematics, the Fraňková–Helly selection theorem is a generalisation of Helly's selection theorem for functions of bounded variation to the case of regulated functions. It was proved in 1991 by the Czech mathematician Dana Fraňková.

Background

Let X be a separable Hilbert space, and let BV(; X) denote the normed vector space of all functions f : → X with finite total variation over the interval , equipped with the total variation norm. It is well known that BV(; X) satisfies the compactness theorem known as Helly's selection theorem: given any sequence of functions (fn)n∈N in BV(; X) that is uniformly bounded in the total variation norm, there exists a subsequence

( f n ( k ) ) ⊆ ( f n ) ⊂ B V ( [ 0 , T ] ; X ) {\displaystyle \left(f_{n(k)}\right)\subseteq (f_{n})\subset \mathrm {BV} (;X)}

and a limit function f ∈ BV(; X) such that fn(k)(t) converges weakly in X to f(t) for every t ∈ . That is, for every continuous linear functional λ ∈ X*,

λ ( f n ( k ) ( t ) ) → λ ( f ( t ) )  in  R  as  k → ∞ . {\displaystyle \lambda \left(f_{n(k)}(t)\right)\to \lambda (f(t)){\mbox{ in }}\mathbb {R} {\mbox{ as }}k\to \infty .}

Consider now the Banach space Reg(; X) of all regulated functions f : → X, equipped with the supremum norm. Helly's theorem does not hold for the space Reg(; X): a counterexample is given by the sequence

f n ( t ) = sin ⁡ ( n t ) . {\displaystyle f_{n}(t)=\sin(nt).}

One may ask, however, if a weaker selection theorem is true, and the Fraňková–Helly selection theorem is such a result.

Statement of the Fraňková–Helly selection theorem

As before, let X be a separable Hilbert space and let Reg(; X) denote the space of regulated functions f : → X, equipped with the supremum norm. Let (fn)n∈N be a sequence in Reg(; X) satisfying the following condition: for every ε > 0, there exists some Lε > 0 so that each fn may be approximated by a un ∈ BV(; X) satisfying

‖ f n − u n ‖ ∞ < ε {\displaystyle \|f_{n}-u_{n}\|_{\infty }<\varepsilon }

and

| u n ( 0 ) | + V a r ( u n ) ≤ L ε , {\displaystyle |u_{n}(0)|+\mathrm {Var} (u_{n})\leq L_{\varepsilon },}

where |-| denotes the norm in X and Var(u) denotes the variation of u, which is defined to be the supremum

sup Π ∑ j = 1 m | u ( t j ) − u ( t j − 1 ) | {\displaystyle \sup _{\Pi }\sum _{j=1}^{m}|u(t_{j})-u(t_{j-1})|}

over all partitions

Π = { 0 = t 0 < t 1 < ⋯ < t m = T , m ∈ N } {\displaystyle \Pi =\{0=t_{0}<t_{1}<\dots <t_{m}=T,m\in \mathbf {N} \}}

of . Then there exists a subsequence

( f n ( k ) ) ⊆ ( f n ) ⊂ R e g ( [ 0 , T ] ; X ) {\displaystyle \left(f_{n(k)}\right)\subseteq (f_{n})\subset \mathrm {Reg} (;X)}

and a limit function f ∈ Reg(; X) such that fn(k)(t) converges weakly in X to f(t) for every t ∈ . That is, for every continuous linear functional λ ∈ X*,

λ ( f n ( k ) ( t ) ) → λ ( f ( t ) )  in  R  as  k → ∞ . {\displaystyle \lambda \left(f_{n(k)}(t)\right)\to \lambda (f(t)){\mbox{ in }}\mathbb {R} {\mbox{ as }}k\to \infty .}

References

Categories: